Using H-Convergence to Calculate the Numerical Errors for 1D Unsaturated Seepage Under Steady-State Conditions

Vista sencilla de ítem
dc.contributor.affiliationDepartment of Civil, Geological and Mining Engineering, Polytechnique Montréal, Montreal, QC, Canada
dc.contributor.affiliationSNC-Lavalin, Montreal, QC, Canada
dc.contributor.affiliationDepartment of Construction Engineering, École de Technologie Supérieure, Montreal, QC, Canada
dc.contributor.affiliationGeotechnical Engineering Team Lead, BBA, Montreal, QC, Canada
dc.contributor.affiliationCollege of Construction Engineering, Jilin University, Changchun, China
dc.contributor.affiliationEnvironmental Engineering program, University of Medellin, Medellin, Colombia
dc.contributor.affiliationDepartment of Mechanical Engineering, Polytechnique Montréal, Montreal, QC, Canada
dc.contributor.authorChapuis R.P.
dc.contributor.authorTaveau C.
dc.contributor.authorDuhaime F.
dc.contributor.authorWeber S.
dc.contributor.authorMarefat V.
dc.contributor.authorZhang L., Blessent D.
dc.contributor.authorBouaanani N., Pelletier D.
dc.date.accessioned2025-12-03T19:34:52Z
dc.date.available2025-12-03T19:34:52Z
dc.date.issued2025
dc.descriptionUnsaturated zones are important for geotechnical design, geochemical reactions, and microbial reactions. The numerical analysis of unsaturated seepage is complex because it involves highly nonlinear partial differential equations. The permeability can vary by orders of magnitude over short vertical distances. This article defines and uses H-convergence tests to quantify numerical errors made by uniform meshes with element size (ES) for 1D steady-state conditions. The quantitative H-convergence should not be confused with a qualitative mesh sensitivity study. The difference between numerical and mathematical convergences is stated. A detailed affordable method for an H-convergence test is presented. The true but unknown solution is defined as the asymptote of the numerical solutions for all solution components when ES decreases to zero. The numerical errors versus ES are then assessed with respect to the true solution, and using a log–log plot, which indicates whether a code is correct or incorrect. If a code is correct, its results follow the rules of mathematical convergence in a mathematical convergence domain (MCD) which is smaller than the numerical convergence domain (NCD). If a code is incorrect, it has an NCD but no MCD. Incorrect algorithms of incorrect codes need to be modified and repaired. Existing codes are shown to converge numerically within large NCDs but generate large errors, up to 500%, in the NCDs, a dangerous situation for designers. © 2024 John Wiley & Sons Ltd.
dc.identifier.doi10.1002/nag.3876
dc.identifier.instnameinstname:Universidad de Medellínspa
dc.identifier.issn3639061
dc.identifier.reponamereponame:Repositorio Institucional Universidad de Medellínspa
dc.identifier.repourlrepourl:https://repository.udem.edu.co/
dc.identifier.urihttp://hdl.handle.net/11407/9267
dc.language.isoeng
dc.publisherJohn Wiley and Sons Ltdspa
dc.publisher.facultyFacultad de Ingenieríasspa
dc.publisher.programIngeniería Ambientalspa
dc.relation.isversionofhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85207119111&doi=10.1002%2fnag.3876&partnerID=40&md5=d152fc39695cc5974ab71587ff70b2a4
dc.relation.referencesBarry, D.A., Parlange, J.Y., Sander, G.C., Sivaplan, M., A Class of Exact Solutions for Richards' equation (1993) Journal of Hydrology, 142 (1-4), pp. 29-46. , https://doi.org/10.1016/0022-1694(93)900003-R
dc.relation.referencesHuang, R.Q., Wu, L.Z., Analytical Solutions to 1-D Horizontal and Vertical Water Infiltration in Saturated/Unsaturated Soils Considering Time-Varying Rainfall (2012) Computers and Geotechnics, 39, pp. 66-72. , https://doi.org/10.1016/j.compgeo.2011.08.008
dc.relation.referencesCao, W., Zhou, A., Shen, S.L., Stability of Unsaturated Soil Slope Considering Stratigraphic Uncertainty and Rotated Anisotropy of Soil Properties (2024) International Journal for Numerical and Analytical Methods in Geomechanics, 48, pp. 359-376. , https://doi.org/10.1002/nag.3642
dc.relation.referencesHuang, S., Tao, R., Wang, R., One Simplified Method for Seismic Stability Analysis of an Unsaturated Slope Considering Seismic Amplification Effect (2023) Geological Journal, 58, pp. 2388-2402. , https://doi.org/10.1002/gj.4769
dc.relation.referencesZhan, L.T., Jia, G.W., Chen, Y.M., Fredlund, D.G., Li, H., An Analytical Solution for Rainfall Infiltration Into an Unsaturated Infinite Slope and Its Application to Slope Stability Analysis (2013) International Journal for Numerical and Analytical Methods in Geomechanics, 37, pp. 1737-1760. , https://doi.org/10.1002/nag.2106
dc.relation.referencesWang, L., Hu, W., Gao, Y., Zhou, E.Q., Li, J., Chen, G.X., Estimation of 3D Earth Pressure With Inclined Backfill Surface Considering Seismic Loads and Suction Effects (2023) International Journal for Numerical and Analytical Methods in Geomechanics, 47, pp. 1086-1099. , https://doi.org/10.1002/nag.3499
dc.relation.referencesChen, B.M., Liao, H., Liu, L.L., Yang, X.L., Bearing Capacity Estimation for Strip Foundation Under Transient Flow (2024) International Journal of Geomechanics, 24 (2). , https://doi.org/10.1061/IJGNAI.GMENG-8557
dc.relation.referencesTang, Z.J., Wang, L., Zhou, A., Sun, D., Li, J., Semi-Analytical Solutions for the Consolidation of Unsaturated Composite Ground With Floating Permeable Columns Under Equal Strain Condition (2024) International Journal for Numerical and Analytical Methods in Geomechanics, 48, pp. 154-183. , https://doi.org/10.1002/nag.3633
dc.relation.referencesZhao, Y.R., Li, X.J., Tai, P., Huang, L.P., Pu, B., Chen, R., Upper Bound Limit Analysis of Roof Collapse of Deep Cavities in Unsaturated Soils (2022) International Journal for Numerical and Analytical Methods in Geomechanics, 46, pp. 1224-1240. , https://doi.org/10.1002/nag.3343
dc.relation.referencesFu, H., Song, E.X., Tong, R., A Numerical Model for Hydrothermal Transport in Unsaturated Freezing Soils Considering Thermodynamics Equilibrium (2023) International Journal for Numerical and Analytical Methods in Geomechanics, 47, pp. 736-758. , https://doi.org/10.1002/nag.3490
dc.relation.referencesYin, Q., Ando, E., Viggiani, G., Sun, W.C., Freezing-Induced Stiffness and Strength Anisotropy in Freezing Clayey Soil: Theory, Numerical Modeling, and Experimental Validation (2022) International Journal for Numerical and Analytical Methods in Geomechanics, 46, pp. 2087-2114. , https://doi.org/10.1002/nag.3380
dc.relation.referencesYan, H.X., Xie, H.J., Rajabi, H., Gu, X.T., Chen, Y., Analytical Solution for One-Dimensional Steady-State VOCs Diffusion in Simplified Capillary Cover Systems (2023) International Journal for Numerical and Analytical Methods in Geomechanics, 47, pp. 2304-2321. , https://doi.org/10.1002/nag.3557
dc.relation.referencesLiu, J., Chen, Z.L., Yu, C., Wang, S., Wu, W., Effect of Retardation and Capillarity on Organic Contaminant Diffusion Through an Unsaturated Composite Liner: An Analytical Approach (2023) International Journal for Numerical and Analytical Methods in Geomechanics, 47, pp. 2096-2115. , https://doi.org/10.1002/nag.3553
dc.relation.referencesThwaites, B., Approximate Calculation of the Laminar Boundary Layer (1949) Aeronaut Quart, 1, pp. 245-280
dc.relation.referencesRoache, P.J., Perspective: A Method for Uniform Reporting of Grid Refinement Studies (1994) ASME Journal of Fluids Engineering, 116 (3), pp. 405-413. , https://doi.org/10.1115/1.2910291
dc.relation.referencesRichardson, L.F., Gaunt, J.A., The Deferred Approach to the Limit. Part I. Single Lattice. Part II. Interpenetrating Lattices (1927) Philosophical Transactions of the Royal Society of London. Series A, 226, pp. 299-361
dc.relation.referencesStern, F., Wilson, R.V., Coleman, H.W., Paterson, E.G., Comprehensive Approach to Verification and Validation of CFD Simulations Part 1: Methodology and Procedures (2001) Journal of Fluids Engineering, 123 (4), pp. 793-802. , https://doi.org/10.1115/1.1412235
dc.relation.referencesChapuis, R.P., Controlling the Quality of Ground Water Parameters: Some Examples (1995) Canadian Geotechnical Journal, 32 (1), pp. 172-177. , https://doi.org/10.1139/t95-014
dc.relation.referencesChapuis, R.P., Duhaime, F., Weber, S., Numerical Convergence Does Not Mean Mathematical Convergence: Examples of Simple Saturated Steady-State Groundwater Models With Pumping Wells (2023) Computers and Geotechnics, 162. , https://doi.org/10.1016/j.compgeo.2023.105615
dc.relation.referencesChapuis, R.P., Numerical Modelling of Reservoirs or Pipes in Groundwater Seepage (2009) Computers and Geotechnics, 36 (5), pp. 895-901. , https://doi.org/10.1016/j.compgeo.2009.01.005
dc.relation.referencesStern, F., Wilson, R.V., Coleman, H.W., Paterson, E.G., (1999) Verification and Validation of CFD Simulations, , Iowa City, IA, Iowa Institute of Hydraulic Research, College of Engineering, The University of Iowa,). IIHR
dc.relation.referencesRoache, P.J., (2009) Fundamentals of Verification and Validation, , Socorro, NM, Hermosa Publishers
dc.relation.referencesChapuis, R.P., Chenaf, D., Acevedo, N., Marcotte, D., Chouteau, M., Unusual Drawdown Curves for a Pumping Test in an Unconfined Aquifer at Lachenaie, Quebec: Field Data and Numerical Modeling (2005) Canadian Geotechnical Journal, 42 (4), pp. 1133-1144. , https://doi.org/10.1139/T05-046
dc.relation.referencesChesnaux, R., Molson, J., Chapuis, R.P., An Analytical Solution for Ground Water Transit Time Through Unconfined Aquifers (2005) Ground Water, 43 (4), pp. 511-517. , https://doi.org/10.1111/j.1745-65842005.0056.x
dc.relation.referencesChenaf, D., Chapuis, R.P., Seepage Face Height, Water Table Position, and Well Efficiency for Steady state (2007) Ground Water, 45 (2), pp. 168-177. , https://doi.org/10.1111/j.1745.6584.2006.00277.x
dc.relation.referencesChapuis, R.P., Saucier, A., A Leaky Aquifer Below Champlain Sea Clay: Closed–form Solutions for Natural Seepage (2013) Ground Water, 51 (6), pp. 960-967. , https://doi.org/10.1111/gwat.12023
dc.relation.referencesChapuis, R.P., Proof of Multiplicity of Solutions for Groundwater Seepage in Recharged Heterogeneous Unconfined Aquifers (2016) International Journal for Numerical and Analytical Methods in Geomechanics, 40 (14), pp. 1988-2002. , https://doi.org/10.1002/nag.2518
dc.relation.referencesChapuis, R.P., A Simple Reason Explains Why It Is so Difficult to Assess Groundwater Ages and Contamination Ages (2017) Science of the Total Environment, 593-594, pp. 109-115. , https://doi.org/10.1016/j.scitotenv.2017.03.140
dc.relation.referencesZhang, L., Chapuis, R.P., Recovery Test After a Constant-Head Test in a Monitoring Well: Interpretation Methods and New Findings (2019) Engineering Geology, 259. , https://doi.org/10.1016/j.enggeo.2019.105150
dc.relation.referencesZhang, L., Chapuis, R.P., Marefat, V., Numerical Values of Shape Factors for Field Permeability Tests in Unconfined Aquifers (2020) Acta Geotechnica, 15 (5), pp. 1243-1257. , https://doi.org/10.1007/d11440-019-00836-4
dc.relation.referencesRoache, P.J., (1998) Verification and Validation in Computational Science and Engineering, , New Mexico, Hermosa Publishers
dc.relation.referencesBanks, J.W., Aslam, T., Rider, W.J., On Sub-Linear Convergence for Linearly Degenerate Waves in Capturing Schemes (2008) Journal of Computational Physics, 227 (14), pp. 6985-7002. , https://doi.org/10.1016/j.jcp.2008.04.002
dc.relation.referencesColeman, H.W., Stern, F., Di Mascio, A., Campana, E., The Problem With Oscillatory Behavior in Grid Convergence Studies (2001) Journal of Fluids Engineering, 123 (2), pp. 438-439. , https://doi.org/10.1115/1.1362672
dc.relation.referencesMorin, P., Nochetto, R.H., Siebert, K.G., Convergence of Adaptive Finite Element Methods (2002) SIAM Review, 44 (4), pp. 631-658
dc.relation.referencesCelik, I., Li, J., Hu, G., Shaffer, C., Limitations of Richardson Extrapolation and Some Possible Remedies (2005) Journal of Fluids Engineering, 127 (4), pp. 795-805. , https://doi.org/10.1115/1.1949646
dc.relation.referencesTruty, A., Zimmermann, T., Stabilized Mixed Finite Element Formulations for Materially Nonlinear Partially Saturated Two-Phase media (2006) Computer Methods in Applied Mechanics and Engineering, 195 (13-16), pp. 1517-1546. , https://doi.org/10.1016/j.cma.2005.05.044
dc.relation.referencesBrock, J.S., Bounded Numerical Error Estimates for Oscillatory Convergence of Simulation Data 18th AIAA Computational Fluids Dynamics Conference, , 2007
dc.relation.referencesPaper 4091
dc.relation.referencesRoy, C.J., Review of Discretization Error Estimators in Scientific Computing 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition 4–7 Jan. 2010, p. 29. , Orlando, Florida, p
dc.relation.referencesRoy, C.J., Oberkampf, W.L., A Comprehensive Framework for Verification, Validation, and Uncertainty Quantification in Scientific Computing (2011) Computer Methods in Applied Mechanics and Engineering, 200 (25-28), pp. 2131-2144. , https://doi.org/10.1016/j.cma.2011.03.016
dc.relation.referencesChapuis, R.P., Masse, I., Madinier, B., Aubertin, M., A Drainage Column Test for Determining Unsaturated Properties of Coarse Materials (2007) Geotechnical Testing Journal, 30 (2), pp. 83-89. , https://doi.org/10.1520/GTJ100385
dc.relation.referencesChapuis, R.P., Masse, I., Madinier, B., Duhaime, F., Water Retention Curves of Coarse Soils Without Organic Matter: Improved Data for Improved Predictions (2015) Geotechnical Testing Journal, 38 (3), pp. 325-337. , https://doi.org/10.1520/GTJ20130154
dc.relation.referencesFerrandon, J., Les Lois D'écoulement de Filtration [The Seepage Filtration Laws] (1948) Le Génie Civil, 125 (2), pp. 24-28. , (in French)
dc.relation.referencesRichards, L.A., Capillary Conduction of Liquids Through Porous Medium (1931) Physics, 1 (5), pp. 318-333. , https://doi.org/10.1063/1.1745010
dc.relation.referencesBrooks, R.H., Corey, A.T., Hydraulic Properties of Porous Media (1964) Hydrology Paper 3, , Fort Collins, Colorado State University
dc.relation.referencesvan Genuchten, M.T., A Closed–Form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils (1980) Soil Science Society of America Journal, 44 (5), pp. 892-898. , https://doi.org/10.2136/sssaj1980.03615995004400050002x
dc.relation.referencesKosugi, K., Three–Parameter Lognormal Distribution Model for Soil Water Retention (1994) Water Resources Research, 30 (4), pp. 891-901. , https://doi.org/10.1029/93WR02931
dc.relation.referencesFredlund, D.G., Xing, A., Equations for the Soil–Water Characteristic Curve (1994) Canadian Geotechnical Journal, 31 (4), pp. 521-532. , https://doi.org/10.1139/t94-061
dc.relation.referencesAubertin, M., Mbonimpa, M., Bussière, B., Chapuis, R.P., A Model to Predict the Water Retention Curve From Basic Geotechnical Properties (2003) Canadian Geotechnical Journal, 40 (6), pp. 1104-1122. , https://doi.org/10.1139/T03-054
dc.relation.referencesMbonimpa, M., Aubertin, M., Bussière, B., Predicting the Unsaturated Hydraulic Conductivity of Granular Soils From Basic Geotechnical Properties Using the Modified Kovács (MK) Model and Statistical Models (2006) Canadian Geotechnical Journal, 43 (8), pp. 773-787. , https://doi.org/10.1139/T06-044
dc.relation.referencesArya, L.M., Bowman, D.C., Thapa, B.B., Cassel, D.K., Scaling Soil Water Characteristics of Golf Course and Athletic Field Sands From Particle–size Distribution (2008) Soil Science Society of America Journal, 72 (1), pp. 25-32. , https://doi.org/10.2136/sssaj2006.0232
dc.relation.referencesFarthing, M.W., Ogden, F.L., Numerical Solution of Richards' Equation: A Review of Advances and Challenges (2017) Soil Science Society of America Journal, 81 (6), pp. 1257-1269. , https://doi.org/10.2136/sssaj2017.02.0058
dc.relation.referencesBaptiste, N., Chapuis, R.P., What Maximum Permeability Can be Measured With a Monitoring Well? (2015) Engineering Geology, 184, pp. 111-118. , https://doi.org/10.1016/j.enggeo.2014.11.006
dc.relation.referencesDelbar, A., Chapuis, R.P., Tracer Tests in a Straight Uniform Flow: New Equations Considering the Distortion of Flow Lines Around the Well (2021) Journal of Contaminant Hydrology, 239. , https://doi.org/10.1016/j.jcnhyd.2021.103776
dc.relation.referencesChapuis, R.P., Aubertin, M., A Simplified Method to Estimate Saturated and Unsaturated Seepage Through Dikes Under Steady-State Conditions (2001) Canadian Geotechnical Journal, 38 (6), pp. 1321-1328. , https://doi.org/10.1139/cgj-38-6-1321
dc.rights.accesoAll Open Access
dc.rights.accesoBronze
dc.rights.accessrightsinfo:eu-repo/semantics/openAccess
dc.sourceInternational Journal for Numerical and Analytical Methods in Geomechanics
dc.sourceInt. J. Numer. Anal. Methods Geomech.
dc.sourceScopus
dc.subjectGroundwater
dc.subjectMathematical convergence
dc.subjectNumerical analysis
dc.subjectNumerical convergence
dc.subjectNumerical errors
dc.subjectUnsaturated
dc.subjectConvergence of numerical methods
dc.subjectGroundwater geochemistry
dc.subjectNonlinear equations
dc.subjectSeepage
dc.subjectSensitivity analysis
dc.subjectConvergence domains
dc.subjectConvergence test
dc.subjectElement sizes
dc.subjectH convergences
dc.subjectMathematical convergence
dc.subjectNumerical convergence
dc.subjectNumerical errors
dc.subjectSteady-state condition
dc.subjectUnsaturated
dc.subjectUnsaturated seepage
dc.subjectMesh generation
dc.titleUsing H-Convergence to Calculate the Numerical Errors for 1D Unsaturated Seepage Under Steady-State Conditions
dc.typeArticle
dc.type.localArtículospa
dc.type.versioninfo:eu-repo/semantics/publishedVersion

Archivos

Colecciones

Indexados Scopus